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Models of the Small World in the General Case

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Models of the Small World in the General Case Models of the Small World A Review M. E. J. Newman Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 Abstract reach their destination, and Milgram found that it had only taken an average of six steps for a letter to get from Ne- It is believed that almost any pair of people in the world braska to Boston. He concluded, with a somewhat cavalier can be connected to one another by a short chain of inter- disregard for experimental niceties, that six was therefore mediate acquaintances, of typical length about six. This the average number of acquaintances separating the pairs phenomenon, colloquially referred to as the “six degrees of of people involved, and conjectured that a similar separa- separation,” has been the subject of considerable recent in- tion might characterize the relationship of any two people terest within the physics community. This paper provides a in the entire world. This situation has been labeled “six de- short review of the topic. grees of separation” (Guare 1990), a phrase which has since passed into popular folklore. Given the form of Milgram’s experiment, one could be 1 Introduction forgiven for supposing that the figure six is probably not a very accurate one. The experiment certainly contained The United Nations Department of Economic and Social many possible sources of error. However, the general re- Affairs estimates that the population of the world exceeded sult that two randomly chosen human beings can be con- six billion people for the first time on October 12, 1999. nected by only a short chain of intermediate acquaintances There is no doubt that the world of human society has be- has been subsequently verified, and is now widely accepted come quite large in recent times. Nonetheless, people rou- (Korte and Milgram 1970). In the jargon of the field this tinely claim that, global statistics notwithstanding, it’s still result is referred to as the small-world effect. a small world. And in a certain sense they are right. Despite The small-world effect applies to networks other than the enormous number of people on the planet, the struc- networks of friends. Brett Tjaden’s parlor game “The Six ture of social networks—the map of who knows whom—is Degrees of Kevin Bacon” connects any pair of film ac- such that we are all very closely connected to one another tors via a chain of at most eight co-stars (Tjaden and Was- (Kochen 1989, Watts 1999). son 1997). Tom Remes hasdone the same forbaseball play- One of the first quantitative studies of the structure of so- ers who have played on the same team (Remes 1997). With cial networks was performed in the late 1960s by Stanley tongue very firmly in cheek, the New York Times played a Milgram, then at Harvard University (Milgram 1967). He similar game with the the names of those who had tangled performed a simple experiment as follows. He took a num- with Monica Lewinsky (Kirby and Sahre 1998). ber of letters addressed to a stockbroker acquaintance of his All of this however, seems somewhat frivolous. Why in Boston, Massachusetts, and distributed them to a random should a serious scientist care about the structure of so- selection of people in Nebraska. (Evidently, he considered cial networks? The reason is that such networks are Nebraska to be about as far as you could get from Boston, crucially important for communications. Most human in social terms, without falling off the end of the world.) communication—where the word is used in its broad- His instructions were that the letters were to be sent to their est sense—takes place directly between individuals. The addressee (the stockbroker) by passing them from person to spread of news, rumors, jokes, and fashions all take place person, and that, in addition, they could be passed only to by contact between individuals. And a rumor can spread arXiv:cond-mat/0001118v2 [cond-mat.stat-mech] 9 May 2000 someone whom the passer new on a first-name basis. Since from coast to coast far faster over a social network in which it was not likely that the initial recipients of the letters were the average degree of separation is six, than it can over on a first-name basis with a Boston stockbroker, their best one in which the average degree is a hundred, or a million. strategy was to pass their letter to someone whom they felt More importantly still, the spread of disease also occurs by was nearer to the stockbroker in some social sense: perhaps person-to-person contact, and the structure of networks of someone they knew in the financial industry, or a friend in such contacts has a huge impact on the nature of epidemics. Massachusetts. In a highly connected network, this year’s flu—or the HIV A reasonable number of Milgram’s letters did eventually virus—can spread far faster than in a network where the 1 paths between individuals are relatively long. Network N ℓ C Crand In this paper we outline some recent developments in the movie actors 225 226 3.65 0.79 0.00027 theory of social networks, particularly in the characteriza- neural network 282 2.65 0.28 0.05 tion and modeling of networks, and in the modeling of the power grid 4941 18.7 0.08 0.0005 spread of information or disease. Table 1: The number of nodes N, average degree of separa- tion ℓ, and clustering coefficient C, for three real-world net- 2 Random graphs works. The last column is the value which C would take in a random graph with the same size and coordination num- The simplest explanation for the small-world effect uses the ber. idea of a random graph. Suppose there is some number N of people in the world, and on average they each have z 1 acquaintances. This means that there are 2 Nz connections A random graph does not show clustering. In a random between people in the entire population. The number z is graph the probability that two of person A’s friends will called the coordination number of the network. be friends of one another is no greater than the probabil- We can make a very simple model of a social network by ity that two randomly chosen people will be. On the other 1 taking N dots (“nodes” or “vertices”) and drawing 2 Nz hand, clustering has been shown to exist in a number of lines (“edges”) between randomly chosen pairs to repre- real-world networks. One can define a clustering coeffi- sent these connections. Such a network is called a random cient C, which is the average fraction of pairs of neigh- graph (Bollob´as 1985). Random graphs have been studied bors of a node which are also neighbors of each other. In a extensively in the mathematics community, particularly by fully connected network, in which everyone knows every- Erd¨os and R´enyi (1959). It is easy to see that a random one else, C = 1; in a random graph C = z/N, which is graph shows the small-world effect. If a person A on such a very small for a large network. In real-world networks it graph has z neighbors, and each of A’s neighbors also has z has been found that, while C is significantly less than 1, it 2 neighbors, then A has about z second neighbors. Extend- is much greater than O(N −1). In Table 1, we show some 3 4 ing this argument A also has z third neighbors, z fourth values of C calculated by Watts and Strogatz (1998) for neighbors and so on. Most people have between a hundred three different networks: the network of collaborations be- 4 and a thousand acquaintances, so z is already between tween movie actors discussed previously, the neural net- 8 12 about 10 and 10 , which is comparable with the popu- work of the worm C. Elegans, and the Western Power Grid lation of the world. In general the number D of degrees of the United States. We also give the value Crand which of separation which we need to consider in order to reach the clustering coefficient would have on random graphs of all N people in the network (also called the diameter of the same size and coordinationnumber, and in each case the the graph) is given by setting zD = N, which implies that measured value is significantly higher than for the random D = log N/ log z. This logarithmic increase in the number graph, indicating that indeed the graph is clustered. of degrees of separation with the size of the network is typ- In the same table we also show the average distance ℓ ical of the small-world effect. Since log N increases only between pairs of nodes in each of these networks. This is slowly with N, it allows the number of degrees to be quite not the same as the diameter D of the network discussed small even in very large systems. above, which is the maximum distance between nodes, but As an example of this type of behavior, Al- it also scales at most logarithmically with number of nodes bert et al. (1999) studied the properties of the network on random graphs. This is easy to see, since the average of “hyperlinks” between documents on the World Wide distance is strictly less than or equal to the maximum dis- Web. They estimated that, despite the fact there were tance, and so ℓ cannot increase any faster than D.
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